1 (* boolean functions over lists *)
3 include "basics/list.ma".
4 include "basics/sets.ma".
5 include "basics/deqsets.ma".
7 (********* search *********)
9 let rec memb (S:DeqSet) (x:S) (l: list S) on l ≝
12 | cons a tl ⇒ (x == a) ∨ memb S x tl
15 notation < "\memb x l" non associative with precedence 90 for @{'memb $x $l}.
16 interpretation "boolean membership" 'memb a l = (memb ? a l).
18 lemma memb_hd: ∀S,a,l. memb S a (a::l) = true.
19 #S #a #l normalize >(proj2 … (eqb_true S …) (refl S a)) //
22 lemma memb_cons: ∀S,a,b,l.
23 memb S a l = true → memb S a (b::l) = true.
24 #S #a #b #l normalize cases (a==b) normalize //
27 lemma memb_single: ∀S,a,x. memb S a [x] = true → a = x.
28 #S #a #x normalize cases (true_or_false … (a==x)) #H
29 [#_ >(\P H) // |>H normalize #abs @False_ind /2/]
32 lemma memb_append: ∀S,a,l1,l2.
33 memb S a (l1@l2) = true →
34 memb S a l1= true ∨ memb S a l2 = true.
35 #S #a #l1 elim l1 normalize [#l2 #H %2 //]
36 #b #tl #Hind #l2 cases (a==b) normalize /2/
39 lemma memb_append_l1: ∀S,a,l1,l2.
40 memb S a l1= true → memb S a (l1@l2) = true.
41 #S #a #l1 elim l1 normalize
42 [normalize #le #abs @False_ind /2/
43 |#b #tl #Hind #l2 cases (a==b) normalize /2/
47 lemma memb_append_l2: ∀S,a,l1,l2.
48 memb S a l2= true → memb S a (l1@l2) = true.
49 #S #a #l1 elim l1 normalize //
50 #b #tl #Hind #l2 cases (a==b) normalize /2/
53 lemma memb_exists: ∀S,a,l.memb S a l = true →
55 #S #a #l elim l [normalize #abs @False_ind /2/]
56 #b #tl #Hind #H cases (orb_true_l … H)
57 [#eqba @(ex_intro … (nil S)) @(ex_intro … tl) >(\P eqba) //
58 |#mem_tl cases (Hind mem_tl) #l1 * #l2 #eqtl
59 @(ex_intro … (b::l1)) @(ex_intro … l2) >eqtl //
63 lemma not_memb_to_not_eq: ∀S,a,b,l.
64 memb S a l = false → memb S b l = true → a==b = false.
65 #S #a #b #l cases (true_or_false (a==b)) //
66 #eqab >(\P eqab) #H >H #abs @False_ind /2/
69 lemma memb_map: ∀S1,S2,f,a,l. memb S1 a l= true →
70 memb S2 (f a) (map … f l) = true.
71 #S1 #S2 #f #a #l elim l normalize [//]
72 #x #tl #memba cases (true_or_false (a==x))
73 [#eqx >eqx >(\P eqx) >(\b (refl … (f x))) normalize //
74 |#eqx >eqx cases (f a==f x) normalize /2/
78 lemma memb_compose: ∀S1,S2,S3,op,a1,a2,l1,l2.
79 memb S1 a1 l1 = true → memb S2 a2 l2 = true →
80 memb S3 (op a1 a2) (compose S1 S2 S3 op l1 l2) = true.
81 #S1 #S2 #S3 #op #a1 #a2 #l1 elim l1 [normalize //]
82 #x #tl #Hind #l2 #memba1 #memba2 cases (orb_true_l … memba1)
83 [#eqa1 >(\P eqa1) @memb_append_l1 @memb_map //
84 |#membtl @memb_append_l2 @Hind //
88 (**************** unicity test *****************)
90 let rec uniqueb (S:DeqSet) l on l : bool ≝
93 | cons a tl ⇒ notb (memb S a tl) ∧ uniqueb S tl
96 (* unique_append l1 l2 add l1 in fornt of l2, but preserving unicity *)
98 let rec unique_append (S:DeqSet) (l1,l2: list S) on l1 ≝
102 let r ≝ unique_append S tl l2 in
103 if memb S a r then r else a::r
106 axiom unique_append_elim: ∀S:DeqSet.∀P: S → Prop.∀l1,l2.
107 (∀x. memb S x l1 = true → P x) → (∀x. memb S x l2 = true → P x) →
108 ∀x. memb S x (unique_append S l1 l2) = true → P x.
110 lemma unique_append_unique: ∀S,l1,l2. uniqueb S l2 = true →
111 uniqueb S (unique_append S l1 l2) = true.
112 #S #l1 elim l1 normalize // #a #tl #Hind #l2 #uniquel2
113 cases (true_or_false … (memb S a (unique_append S tl l2)))
114 #H >H normalize [@Hind //] >H normalize @Hind //
117 (******************* sublist *******************)
119 λS,l1,l2.∀a. memb S a l1 = true → memb S a l2 = true.
121 lemma sublist_length: ∀S,l1,l2.
122 uniqueb S l1 = true → sublist S l1 l2 → |l1| ≤ |l2|.
124 #a #tl #Hind #l2 #unique #sub
125 cut (∃l3,l4.l2=l3@(a::l4)) [@memb_exists @sub //]
126 * #l3 * #l4 #eql2 >eql2 >length_append normalize
127 applyS le_S_S <length_append @Hind [@(andb_true_r … unique)]
128 >eql2 in sub; #sub #x #membx
129 cases (memb_append … (sub x (orb_true_r2 … membx)))
130 [#membxl3 @memb_append_l1 //
131 |#membxal4 cases (orb_true_l … membxal4)
132 [#eqxa @False_ind lapply (andb_true_l … unique)
133 <(\P eqxa) >membx normalize /2/ |#membxl4 @memb_append_l2 //
138 lemma sublist_unique_append_l1:
139 ∀S,l1,l2. sublist S l1 (unique_append S l1 l2).
140 #S #l1 elim l1 normalize [#l2 #S #abs @False_ind /2/]
142 normalize cases (true_or_false … (a==x)) #eqax >eqax
143 [<(\P eqax) cases (true_or_false (memb S a (unique_append S tl l2)))
144 [#H >H normalize // | #H >H normalize >(\b (refl … a)) //]
145 |cases (memb S x (unique_append S tl l2)) normalize
146 [/2/ |>eqax normalize /2/]
150 lemma sublist_unique_append_l2:
151 ∀S,l1,l2. sublist S l2 (unique_append S l1 l2).
152 #S #l1 elim l1 [normalize //] #x #tl #Hind normalize
153 #l2 #a cases (memb S x (unique_append S tl l2)) normalize
154 [@Hind | cases (a==x) normalize // @Hind]
157 lemma decidable_sublist:∀S,l1,l2.
158 (sublist S l1 l2) ∨ ¬(sublist S l1 l2).
160 [%1 #a normalize in ⊢ (%→?); #abs @False_ind /2/
162 [cases (true_or_false (memb S a l2)) #memba
163 [%1 whd #x #membx cases (orb_true_l … membx)
164 [#eqax >(\P eqax) // |@subtl]
165 |%2 @(not_to_not … (eqnot_to_noteq … true memba)) #H1 @H1 @memb_hd
167 |%2 @(not_to_not … subtl) #H1 #x #H2 @H1 @memb_cons //
172 (********************* filtering *****************)
174 lemma filter_true: ∀S,f,a,l.
175 memb S a (filter S f l) = true → f a = true.
176 #S #f #a #l elim l [normalize #H @False_ind /2/]
177 #b #tl #Hind cases (true_or_false (f b)) #H
178 normalize >H normalize [2:@Hind]
179 cases (true_or_false (a==b)) #eqab
180 [#_ >(\P eqab) // | >eqab normalize @Hind]
183 lemma memb_filter_memb: ∀S,f,a,l.
184 memb S a (filter S f l) = true → memb S a l = true.
185 #S #f #a #l elim l [normalize //]
186 #b #tl #Hind normalize (cases (f b)) normalize
187 cases (a==b) normalize // @Hind
190 lemma memb_filter: ∀S,f,l,x. memb S x (filter ? f l) = true →
191 memb S x l = true ∧ (f x = true).
194 lemma memb_filter_l: ∀S,f,x,l. (f x = true) → memb S x l = true →
195 memb S x (filter ? f l) = true.
196 #S #f #x #l #fx elim l normalize //
197 #b #tl #Hind cases (true_or_false (x==b)) #eqxb
198 [<(\P eqxb) >(\b (refl … x)) >fx normalize >(\b (refl … x)) normalize //
199 |>eqxb cases (f b) normalize [>eqxb normalize @Hind| @Hind]
203 (********************* exists *****************)
205 let rec exists (A:Type[0]) (p:A → bool) (l:list A) on l : bool ≝
208 | cons h t ⇒ orb (p h) (exists A p t)
211 lemma Exists_exists : ∀A,P,l.
214 #A #P #l elim l [ * | #hd #tl #IH * [ #H %{hd} @H | @IH ]